Lower deviations for branching processes with immigration
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929401523863552 |
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| author | Sharipov, Sadillo Wachtel, Vitali |
| author_facet | Sharipov, Sadillo Wachtel, Vitali |
| contents | Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for $Y_{n}$. More precisely, assuming that $k,n \to \infty$ such that $k=o\left(n \right)$, we investigate the asymptotics of $\mathbf{P}\left(Y_{n} \leq k \right)$ and $\mathbf{P}\left(Y_{n} = k \right)$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proven by Mellein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower deviations for branching processes with immigration Sharipov, Sadillo Wachtel, Vitali Probability 60J80 Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for $Y_{n}$. More precisely, assuming that $k,n \to \infty$ such that $k=o\left(n \right)$, we investigate the asymptotics of $\mathbf{P}\left(Y_{n} \leq k \right)$ and $\mathbf{P}\left(Y_{n} = k \right)$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proven by Mellein. |
| title | Lower deviations for branching processes with immigration |
| topic | Probability 60J80 |
| url | https://arxiv.org/abs/2406.18724 |